The radius and boundary singularity conjecture for mirror maps

Let $N_1,N_2,\ldots,N_k\ge2$ be integers with $\Phi_{\mathbf{N}}\ge4$, and let ${\bf z}_{\bf N}(q)$ be the mirror map with associated function ${\bf q}_{\bf N}(z)$ and constant $C_{\bf N}$. Radius and boundary singularity conjecture. The radius of convergence of the Taylor series of ${\bf z}_{\bf N}(q)$ is

$${\bf q}_{\bf N}(1/C_{\bf N}).$$

Moreover, ${\bf q}_{\bf N}(1/C_{\bf N})$ is the only singularity of ${\bf z}_{\bf N}(q)$ on the boundary of its disk of convergence. The conjecture is motivated by extensive numerical calculations for cases with large $\Phi_{\mathbf N}$; the paper discusses a possible proof and its relation to the conjecture on Taylor-coefficient signs.